Mini Tutorial
MT-224
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The Butterworth Response
by Hank Zumbahlen,
Analog Devices, Inc.
IN THIS MINI TUTORIAL
10429-001
The Butterworth filter, a precision op amp based circuit
block, is one of multiple discrete circuits described in a series
of mini tutorials.
INTRODUCTION
The Butterworth filter is the ideal compromise between
attenuation and phase response. Because it has no ripple in the
pass band or the stop band, it is sometimes referred to as
a maximally flat filter. The Butterworth filter achieves its
flatness at the expense of a relatively wide transition region
from pass band to stop band, with average transient
characteristics.
The normalized poles of the Butterworth filter fall on the unit
circle (in the s plane). The pole positions are given by
− sin
2K − 1)π
(2K − 1)π
+ j cos
2n
2n
where:
K = 1, 2,…n and is the pole pair number.
n is the number of poles.
The poles are spaced equidistant on the unit circle, which
means the angles between the poles are equal. Figure 1 shows
the pole placement for a 5-pole Butterworth filter.
Figure 1. Butterworth Pole Location
Given the pole locations, ω0 and α (or Q) can be determined.
These values can then be used to determine the component
values of the filter. The design tables for passive filters use
frequency and impedance normalized filters. They are normalized to a frequency of 1 rad/sec and impedance of 1 Ω. These
filters can be denormalized to determine actual component
values. This allows the comparison of the frequency domain
and/or time domain responses of the various filters on equal
footing. The Butterworth filter is normalized for a –3 dB
response at ω0 = 1.
The values of the elements of the Butterworth filter are more
practical and less critical than many other filter types. The
frequency response, group delay, impulse response, and step
response are shown in Figure 2 through Figure 6. The pole
locations and corresponding ωo and α terms are tabulated
in Table 1.
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FREQUENCY RESPONSE, GROUP DELAY, IMPULSE RESPONSE, AND STEP RESPONSE
The frequency response, group delay, impulse response, and step response, as well as the amplitude, are cataloged in Figure 2 through
Figure 6.
2.0
–90
0.1
0.2
0.4
0.8 1.1
2.0
4.0
1.0
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DELAY (s)
–50
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AMPLITUDE (dB)
0
0
0.1
8.0 10
0.4
FREQUENCY (Hz)
1.1
4.0
10
FREQUENCY (Hz)
Figure 5. Butterworth Response, Group Delay
Figure 2. Butterworth Response, Amplitude
1.0
1.2
0.2
0.4
0.8
1.1
0
0
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AMPLITUDE (V)
4.0
–4.0
3
2
3
4
Figure 6. Butterworth Response, Step Response
8.0
2
1
TIME (s)
Figure 3. Butterworth Response, Amplitude Detail
1
0.4
0
2.0
FREQUENCY (Hz)
0
0.8
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–4.0
0.1
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AMPLITUDE (V)
AMPLITUDE (dB)
0
4
5
TIME (s)
Figure 4. Butterworth Response, Impulse Response
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BUTTERWORTH DESIGN
The pole location and corresponding ω and α terms for these values are tabulated in Table 1.
Table 1.
Order
Section Real Part
Imaginary
Part
F0
α
Q
−3 dB
Frequency
2
1
0.7071
0.7071
1.0000
1.4142
0.7071
1.0000
3
1
0.5000
0.8660
1.0000
1.0000
1.0000
2
1.0000
4
5
6
7
8
9
10
1.0000
Peaking
Frequency
Peaking
Level
0.7071
1.2493
0.8409
3.0102
0.8995
4.6163
0.9306
6.0210
0.4717
0.2204
0.9492
7.2530
1.0000
1
0.9239
0.3827
1.0000
1.8478
0.5412
2
0.3827
0.9239
1.0000
0.7654
1.3065
1
0.8090
0.5878
1.0000
1.6180
0.6180
2
0.3090
0.9511
1.0000
0.6180
1.6182
3
1.0000
1
0.9659
0.2588
1.0000
1.9319
0.5176
0.6758
2
0.7071
0.7071
1.0000
1.4142
0.7071
1.0000
3
0.2588
0.9659
1.0000
0.5176
1.9319
1
0.9010
0.4339
1.0000
1.8019
0.5550
2
0.6235
0.7818
1.0000
1.2470
0.8019
3
0.2225
0.9749
1.0000
0.4450
2.2471
4
1.0000
1
0.9808
0.1951
1.0000
1.9616
0.5098
0.6615
2
0.8315
0.5556
1.0000
1.6629
0.6013
0.8295
3
0.5556
0.8315
1.0000
1.1112
0.9000
0.6186
0.6876
4
0.1951
0.9808
1.0000
0.3902
2.5628
0.9612
8.3429
1.0000
0.7195
0.8588
1.0000
1.0000
0.7449
1.0000
1
0.9397
0.3420
1.0000
1.8794
0.5321
0.7026
2
0.7660
0.6428
1.0000
1.5320
0.6527
0.9172
3
0.5000
0.8660
1.0000
1.0000
1.0000
0.7071
1.2493
4
0.1737
0.9848
1.0000
0.3474
2.8785
0.9694
9.3165
5
1.0000
1
0.9877
0.1564
1.0000
1.9754
0.5062
0.6549
2
0.8910
0.4540
1.0000
1.7820
0.5612
0.7564
3
0.7071
0.7071
1.0000
1.4142
0.7071
1.0000
4
0.4540
0.8910
1.0000
0.9080
1.1013
0.7667
1.8407
5
0.1564
0.9877
1.0000
0.3128
3.1970
0.9752
10.2023
1.0000
1.0000
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REFERENCES
Zumbahlen, Hank. Linear Circuit Design Handbook. Elsevier. 2008. ISBN: 978-7506-8703-4.
REVISION HISTORY
4/12—Revision 0: Initial Version
©2012 Analog Devices, Inc. All rights reserved. Trademarks and
registered trademarks are the property of their respective owners.
MT10429-0-4/12(0)
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